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Physical Review A

Phase fluctuations in a laser with atomic memory effects

Miguel Orszag, C. Saavedra

Published Jan 1, 1991• 22 citations
Publisher

Abstract

We calculate phase fluctuations of the phase operator defined by Pegg and Barnett [Europhys. Lett. 6, 483 (1988); J. Mod. Opt. 36, 7 (1989); Phys. Rev. A 39, 1665 (1989)] in a laser with atomic memory effects. First, we compute the phase fluctuations of a laser, using the Scully-Lamb theory. In a single curve of 〈(\ensuremath{\Delta}\ensuremath{\varphi}${)}^{2}$〉 versus time, we obtain the expected behavior: namely, for Dt\ensuremath{\sim}1 the Schawlow-Townes linewidth and for t\ensuremath{\gg}${\mathit{D}}^{\mathrm{\ensuremath{-}}1}$ the phase fluctuations corresponding to a random phase. As we include the atomic memory effects, we obtain a noise reduction for times of the order of the atomic decay time.

DOI
10.1103/physreva.43.554

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We calculate phase fluctuations of the phase operator defined by Pegg and Barnett [Europhys. Lett. 6, 483 (1988); J. Mod. Opt. 36, 7 (1989); Phys. Rev. A 39, 1665 (1989)] in a laser with atomic memory effects. First, we compute the phase fluctuations of a laser, using the Scully-Lamb theory. In a single curve of 〈(\ensuremath{\Delta}\ensuremath{\varphi})2{)}^{2})2〉 versus time, we obtain the expected behavior: namely, for Dt\ensuremath{\sim}1 the Schawlow-Townes linewidth and for t\ensuremath{\gg}D\ensuremath−1{\mathit{D}}^{\mathrm{\ensuremath{-}}1}D\ensuremath−1 the phase fluctuations corresponding to a random phase. As we include the atomic memory effects, we obtain a noise reduction for times of the order of the atomic decay time.
We calculate the phase-difference and phase-sum fluctuations of the quantum-beat laser, using the nonlinear model of Bergou et al. [Phys. Rev. A 38, 754 (1988)], and also making use of the phase operators defined by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)]. We found a vanishing phase-difference diffusion constant, in agreement with Bergou et al. However, the present formalism is more general, and includes shot noise as well as saturation effects for the relative and sum phase fluctuations.
In papers by Lynch [ Phys. Rev. A 41 2841 (1990)] and Gerry and Urbanski [ Phys. Rev. A 42 662 (1990)] it has been argued that the phase-fluctuation laser experiments of Gerhardt, Büchler and Lifkin [ Phys. Lett. 49A 119 (1974)] are in good agreement with the variance of the Pegg–Barnett phase operator for a coherent state, even for a small number of photons. We argue that this is not conclusive. In fact, we show that the variance of the phase in fact depends on the relative phase between the phase of the coherent state and the off-set phase ϕ 0 of the Pegg–Barnett phase operator. This off-set phase is replaced with the phase of a reference beam in an actual experiment and we show that several choices of such a relative phase can be fitted to the experimental data. We also discuss the Noh, Fougères and Mandel [ Phys. Rev. A 46 2840 (1992)] relative phase experiment in terms of the Pegg–Barnett phase taking post-selection conditions into account.
The linear theory of the correlated-emission laser (CEL) is studied using the quantum-mechanical definition of the phase of the electromagnetic field recently introduced by Barnett and Pegg [J. Mod. Opt. 36, 17 (1989); Phys. Rev. A 39, 1665 (1989)]. From the master equation for the density matrix of the field, a Fokker-Planck equation for the phase probability function is derived, and the phase properties of the CEL discussed. The Adler equation and the equation of motion for the variance of the phase difference are derived. The Fokker-Planck equation satisfied by the phase probability function is compared with the corresponding equations for the P, Q, and Wigner quasiprobability distributions, specialized to the case when amplitude fluctuations are negligible. We show that the phase probability function and the Wigner function satisfy the same equation, and conclude that, of the three quasiprobability functions studied, only the Wigner function is reliable for calculating directly the phase properties. The commonly used P function overestimates the laser linewidth by a factor of 2.
The expectation value and variance of a new phase operator introduced recently by Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and by Barnett and Pegg [J. Mod. Opt. 36, 7 (1989)] is calculated for a coherent state. The values for the variance so obtained are compared with the phase-fluctuation measurements of Gerhardt, B\"uchler, and Liftin [Phys. Lett. 49A, 119 (1974)] and Gerhardt, Welling, and Frolich [Appl. Phys. 2, 91 (1973)]. Agreement is found to be good, provided that account is taken of the details of the experimental values.
The question of how the light output of a laser and its fluctuations depend on the optical-cavity length at a given frequency is investigated theoretically for a laser oscillating in a single Gaussian mode. The treatment is quantum mechanical, and is based on the Scully-Lamb laser model, except that a perturbation expansion is used and possible cooperative atomic effects are included. It is shown, with the help of some reasonable approximations, that the probability distribution of the photon-occupation number can be cast into a form that is similar to the Scully-Lamb formula, but with coefficients that depend on cavity length in a more complicated way. Curves are presented that illustrate the behavi\`or and should lend themselves to experimental test.